In (REF ), blackhowever, only the quantile is dynamic, while the factor \(1+e^{\gamma _{0j}}\)  remains constant through time. Therefore, in order to generalize this approach, we also consider the alternative formulation for the ES presented in [1]}, where the difference between the ES and the VaR is modeled using an AutoRegressive (AR) specification as follows:
\(ES_{tj} &= {\cal Q}_{Y_{tj}}(\tau _j) - x_{tj}, \quad \quad t=1,2,...,T, \quad j=1,2,...,p, \\x_{tj} &= (\gamma _{1j} + \gamma _{2j} ({\cal Q}_{Y_{t-1j}}(\tau _j) - Y_{t-1j}) + \gamma _{3j} x_{t-1j}) 1_{(Y_{tj} \le {\cal Q}_{Y_{tj}})} + x_{t-1j} 1_{(Y_{tj} > {\cal Q}_{Y_{tj}})},\) 
